There are several state-of-the-art filters, including optical filters, for use in optical telecommunications networks and the like. These include Butterworth filters, Chebychev filters, Bessel filters, and elliptical filters, among others.
A Butterworth filter is characterized by a response that is relatively flat within its pass band, but which cuts off relatively sharply outside of its pass band. It is essentially a low-pass filter with its poles arranged with equal spacing around a semicircular locus. Thus, the Butterworth filter represents a maximally flat transmission filter.
As is well known to those of ordinary skill in the art, by bringing the poles closer to the jω axis (and thereby increasing their quality factors (Qs)) in equal proportion such that the poles lie on an ellipse, the resulting filter has a frequency cutoff that is steeper than that of the Butterworth filter. However, the effects of each pole are visible in the frequency response of the filter, resulting in a variation in amplitude that is known as ripple in the pass band. With proper pole arrangement, variations can be made equal, providing a Chebychev filter. Thus, the Chebychev filter represents an equal transmission ripple filter.
The Bessel filter represents a tradeoff in the opposite direction. The Bessel filter's poles lie on a locus that is further from the jω axis, increasing transient response, but at the expense of a less steep cutoff in the stop band. Thus, the Bessel filter represents a flat group delay filter. The Bessel filter also represents a maximally linear phase filter, within a certain pass band around its center frequency.
By increasing the Qs of the poles nearest the pass band edge, a filter with shaper stop band cutoff than that of the Chebychev filter is obtained, without incurring more pass band ripple. The resulting gain peak is compensated for by providing a zero at the bottom of the stop band, and additional zeroes are spaced along the stop band to ensure that the filter response remains below a predetermined level of stop band attenuation. The result is the elliptical filter. Thus, the elliptical filter represents an equal rejection ripple filter. The elliptical filter's high-Q poles produce a transient response that is worse than that of the Chebychev filter. In addition, the elliptical filter is not realizable as an optical filter in the form of a multi-layer dielectric interference optical filter, as its transfer function has a numerator polynomial, generating non-infinity transmission zeroes. The first three filter types are realizable as optical filters, including multi-layer dielectric interference optical filters, as their transfer functions have only denominator polynomials.
In general, the Butterworth filter has desirable maximally flat transmission within its pass band, which gradually rolls off outside of the pass band. The Chebychev filter has significantly steeper roll off outside of its pass band, but its group delay is more severe than that of the Butterworth filter due to the extra group delay ripple associated with the equal transmission ripple within the pass band. The phase non-linearity of the Chebychev filter is also more severe than that of the Butterworth filter because of the phase non-linearity associated with the equal transmission ripple within the pass band. The Bessel filter has maximally flat group delay and maximally linear phase within its pass band, but rolls off even more gradually than the Butterworth filter outside of the pass band. Each of these filters is realizable as a multi-coupled resonant structure, such as an optical ring resonator, a Fabry-Perot resonator, or the like.
What is still needed in the art, however, is a filter that has its group delay limited between two boundaries, an upper boundary and a lower boundary. Thus, the peak-to-peak group delay ripple of the filter would not exceed the difference between the two boundaries. What is also still needed in the art is a filter that has its phase non-linearity limited between two boundaries, an upper boundary and a lower boundary. Thus, the phase of the filter would have minimum deviation from linear, not exceeding half the difference between the two boundaries. The eye closure penalty caused by the dispersion of optical filters is proportional to the amount of deviation of their phase characteristics from linear within the spectral band of the signal. The reduction of the deviation of these phase characteristics from linear within the spectral band of the signal is essential to the reduction of the eye closure penalty caused by the dispersive properties of such optical filters.